I think some of them must. I have had 3 parakeets and all of them have had some sort of problems. They are otherwise healthy birds and live pretty happy lives, but I really wonder because of the congenital defects they have.

My pair that I have now, the boy bird is very tiny and missing one claw, like it never grew, ever. He has a little less grip when landing because of that, but he is otherwise fine.

The girl's feathers on one wing don't fold correctly and she can't fly very well at all. She can't get more than a few inches off the ground and to get back into the cage she has to climb up on the couch so she is on the same level with it, sometimes she still doesn't make it the first time.

It just really makes me wonder if the breeders don't get birds from elsewhere often enough and they are just getting little defects from inbreeding.

Most animals in captivity have inbreed problems.
Budgie parakeets for example, there is only one color of them in the wild - green. Basically all the other colors we see are the result of selective breeding. In other words, inbreeding.

Inbreeding might not cause defects, but in theory it can certainly produce weaker offspring which can easily catch diseases and become sick.

In order to breed a certain trait, a lot of people are still using selective breed to produce the results they want. So yes, inbreeding happens a lot for pet birds and many other pets as well.

I think inbred is a bit hard to avoid since most pets live together. One way to reduce the likelihood of this is by neutering which night be too drastic especially in one desires more progeny. The other method is to keep the birds in one acid but composing of different age groups.